On the existence of at least a solution for functional integral equations via measure of noncompactness

Abstract

In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation

$$u\left(t\right)=g(t,u(t\left)\right)+{\int }_0^{t}G(t,s,u(s\left)\right)ds,t\in [0,+\infty [,$$ in the space of all bounded and continuous real functions on ${\mathbb{R}}_{+}$, under suitable assumptions on $g$ and $G$. Also, we establish an extension of Darbo’s fixed-point theorem and discuss some consequences.